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Disagreements about the somatic risks from low doses of ionizing radiation stem from two difficulties fundamental to the logic of inference from observational data. First, precise direct estimation of small risks requires impracticably large samples. Second, precise estimates of low-dose risks based largely on high-dose data, for which the sample size requirements are more easily satisfied, must depend heavily on assumptions about the shape of the dose-response curve, even when only a few of the parameters of the theoretical form of the curve are unknown.